Question 32
For a non-negative integer $n$ the $n$-th iterate of a function $f: \mathbb{R} \to \mathbb{R}$ is $f^n = \underbrace{f \circ \dots \circ f}_{n \text{ times}}$, and $f^0$ is the identity function. Determine the continuous functions $f: \mathbb{R} \to \mathbb{R}$, that satisfy simultaneously the conditions:
a) The function $f^0 + f^1$ is increasing.
b) There is a positive integer $m$ such that the function $f^0 + \dots + f^m$ is decreasing.
a) The function $f^0 + f^1$ is increasing.
b) There is a positive integer $m$ such that the function $f^0 + \dots + f^m$ is decreasing.